This paper presents a distributed, linear-time algorithm for localizing sensor network nodes in the presence of range measurement noise. The algorithm uses the probabilistic notion of robust quadrilaterals to avoid flip ambiguities that can corrupt localization computations. The localization problem is formulated as a two-dimensional graph realization problem, where the goal is to recover the Euclidean position of each vertex up to a global rotation and translation. This approach is applicable to sensor networks where each node can estimate distances to its neighbors but lacks absolute position references like GPS or fixed anchors. The algorithm is implemented on a physical sensor network and empirically assessed for accuracy and performance. Simulations show that the algorithm scales to large networks and handles real-world deployment geometries. It also supports the localization of mobile nodes. Key characteristics of the algorithm include support for noisy distance measurements, full distribution without beacons or anchors, and high probability of correctly localizing each node or not at all. Cluster-based localization allows dynamic node insertion and mobility. The paper discusses related work, including theoretical foundations for network localization, error analysis, and previous approaches to localization using global optimization or propagation of location information from known reference nodes. The algorithm uses robust quadrilaterals as a building block for localization, adding an additional constraint beyond graph rigidity. This constraint minimizes the probability of realizing flip ambiguities due to measurement noise. The algorithm is broken into three phases: cluster localization, cluster optimization (optional), and cluster transformation. Each phase contributes to the overall localization process, with cluster localization being the first step where clusters are localized into local coordinate systems. The algorithm's robustness is proven through analysis of error propagation and the worst-case probability of error. It is shown that robust quadrilaterals significantly reduce error propagation compared to basic trilateration methods. The algorithm is tested on both physical and simulated networks, demonstrating its effectiveness in real-world scenarios and scalability to large networks. The results show that the algorithm performs well in terms of localization accuracy and can handle mobile nodes effectively.This paper presents a distributed, linear-time algorithm for localizing sensor network nodes in the presence of range measurement noise. The algorithm uses the probabilistic notion of robust quadrilaterals to avoid flip ambiguities that can corrupt localization computations. The localization problem is formulated as a two-dimensional graph realization problem, where the goal is to recover the Euclidean position of each vertex up to a global rotation and translation. This approach is applicable to sensor networks where each node can estimate distances to its neighbors but lacks absolute position references like GPS or fixed anchors. The algorithm is implemented on a physical sensor network and empirically assessed for accuracy and performance. Simulations show that the algorithm scales to large networks and handles real-world deployment geometries. It also supports the localization of mobile nodes. Key characteristics of the algorithm include support for noisy distance measurements, full distribution without beacons or anchors, and high probability of correctly localizing each node or not at all. Cluster-based localization allows dynamic node insertion and mobility. The paper discusses related work, including theoretical foundations for network localization, error analysis, and previous approaches to localization using global optimization or propagation of location information from known reference nodes. The algorithm uses robust quadrilaterals as a building block for localization, adding an additional constraint beyond graph rigidity. This constraint minimizes the probability of realizing flip ambiguities due to measurement noise. The algorithm is broken into three phases: cluster localization, cluster optimization (optional), and cluster transformation. Each phase contributes to the overall localization process, with cluster localization being the first step where clusters are localized into local coordinate systems. The algorithm's robustness is proven through analysis of error propagation and the worst-case probability of error. It is shown that robust quadrilaterals significantly reduce error propagation compared to basic trilateration methods. The algorithm is tested on both physical and simulated networks, demonstrating its effectiveness in real-world scenarios and scalability to large networks. The results show that the algorithm performs well in terms of localization accuracy and can handle mobile nodes effectively.